A generalized Trudinger–Moser inequality on a compact Riemannian surface. (April 2018) - Zhu, Xiaobao
Abstract: Let ( Σ, g ) be a compact Riemannian surface. Let ψ, h be two smooth functions on Σ with ∫ Σ ψ d v g ≠ 0 and h ≥ 0, h ≢ 0 . In this paper, using a method of blow-up analysis, we prove that the functional (1) J ψ, h ( u ) = 1 2 ∫ Σ | ∇ g u | 2 d v g + 8 π 1 ∫ Σ ψ d v g ∫ Σ ψ u d v g − 8 π log ∫ Σ h e u d v g is bounded from below in W 1, 2 ( Σ, g ) . Moreover, we obtain a sufficient condition under which J ψ, h attains its infimum in W 1, 2 ( Σ, g ) . These results generalize the main results in Ding–Jost–Li–Wang (1997) and Yang–Zhu (2017).